TRANSIENT OSCILLATIONS IN ELECTRIC WAVE-FILTERS 17 



IV. The Building-Up of Alternating Currents in Wave- 

 Filters 



If an e.m.f. sin (wt+Q) is applied to the low pass wave-filter (type 

 Lid) at time t = 0, then by formulas I and (la), the resultant current 

 in the «th section builds up in accordance with the expression 



A s[nQlj2n(xi)cos\(x — Xi)dxi-\-cosQjj2n(xi)s[n\(x-Xi)dxi , 



where x = w c t and X = «/w c . 



For the band pass filter, type LiCiL 2 C 2 , the corresponding formula, 

 based on the approximations discussed in the preceding, is by I and 



(3a). 



j^j\n(ny+G)j J 2 n(yi) cos (X-ju) (y-yi)dyi 



+ cos {ixy + Q)j Jiniyi) sin (X-/t) (y-yi)dyi~\ 



where y = wt/2; \ = 2oo/w; and \x = 2u m lw so that fxy = co m t. Similar 

 formulas are deducible for the other types of band pass filters con- 

 sidered in the preceding section. 



Comparison of these formulas shows that, in both the low pass and 

 band pass wave-filters, the genesis and growth of the current in re- 

 sponse to an e.m.f. sin (w£+9), applied at time / = 0, is mathematically 

 determined by definite integrals of the form 



and 



S„(z; p)= I Jn(zi) sin v(z-Zi)dz u 

 Jo 



C n (z; v) = / J n (zi) cos v(z-z{)dzi. 



These integrals 13 have been extensively studied in the course of 

 this investigation; their general properties and the appropriate meth- 

 ods of computation are discussed in Appendix III. 



The subsidence of the current, when a sinusoidal e.m.f. is removed, 

 is also determined by the above formulas for the low pass and band 

 pass filters. To show this suppose that prior to the reference time 

 t = 0, that steady-state currents are flowing in the filter in response 



IS The writers take pleasure in acknowledging their indebtedness to T. H. Gronwall, 

 consulting mathematician, who furnished asymptotic formulas for the computation 

 of these integrals. See Appendix III. 



